How do I curve fit the data set

Hello all,

I'm facing difficulty in fitting the data.

The red ones are from the data set, and I want to fit that as like blue one (Experimental Data). How can I do that. I have also attached the data set for your reference.

Thank You.

[Sam Chak]

Hi @Prajwal Magadi

What is the mathematical function of the blue curve? Is it a skewed normal distribution function?

Can you also suggest some candidate functions for fitting into the red data? Look up some Kernel functions.

Red data seems to have discontinuities at multiple intervals. Is it acceptable to have a piecewise function to fit the data?

[Prajwal Magadi]

The blue curve is also a data set from an experimental obervation.

The red data is from the simulation.

And I have no idea of the function type of blue data.

Answer 1

[Sam Chak]

Hi @Prajwal Magadi

Not sure if this is what your want. But you can try finding the best math function to fit.

data = load('curve_fit.mat');
x    = data.theta_degree';
y    = data.x';

skewEqn = 'a/(sqrt(2*pi))*exp(- b*(x - c)^2)*((1/2)*(1 + erf(e*(x - c)/sqrt(2)))) + d';

fo = fitoptions('Method', 'NonlinearLeastSquares',...
               'Lower', [  0, 0,  5,  0, 0.1],...   % {a, b, c, d, e}
               'Upper', [100, 1, 20, 10, 1.0],...
               'StartPoint', [50 0.5 10 5 0.5]);

ft = fittype(skewEqn, 'options', fo);

[yfit, gof] = fit(x, y, ft)

yfit = 
     General model:
     yfit(x) = a/(sqrt(2*pi))*exp(- b*(x - c)^2)*((1/2)*(1 + erf(e*(x - c)/sqrt(2)
                    ))) + d
     Coefficients (with 95% confidence bounds):
       a =       85.15  (84.38, 85.93)
       b =    0.003494  (0.003423, 0.003564)
       c =       7.162  (7.116, 7.209)
       d =       2.461  (2.241, 2.681)
       e =      0.3845  (0.3752, 0.3937)

gof = struct with fields:
           sse: 9.7227e+04
       rsquare: 0.9063
           dfe: 9995
    adjrsquare: 0.9062
          rmse: 3.1189

plot(yfit, x,  y)
grid on, xlabel('\theta'), ylabel('x')
legend('Data', 'Fitted Skew Dist Fcn')

[Prajwal Magadi]

Thank you Sam.

[Sam Chak]

@Prajwal Magadi, Don't mention it. Have a nice weekend!

Answer 2

[Alex Sha]

@Prajwal Magadi, one more function:

Sum Squared Error (SSE): 75571.6557870726
Root of Mean Square Error (RMSE): 2.74902993412354
Correlation Coef. (R): 0.962878352853849
R-Square: 0.927134722394541

Parameter	Best Estimate    
---------	-------------    
y0       	3.57509887406416 
a        	10210313.7484567 
xc       	17.1109591760412 
w1       	18.7010167618592 
w2       	-1.38399464317597
w3       	-1.49504622174935

[Sam Chak]

Nice one, @Alex Sha. 👍

Can you advise how you selected the candidate as the product of the logistic function and its scaled reflection?

By the experience of seeing hundreds of different single-hump skewed distribution functions?

x  = linspace(-10, 20, 30001);
y1 = 1./(1 + exp(- (x - 1)));
y2 = 1 - 1./(1 + exp(- (x - 1)/5));
y  = y1.*y2;
plot(x, y1, 'linewidth', 2, 'color', '#7c98c3'), hold on
plot(x, y2, 'linewidth', 2, 'color', '#e5ab45'), 
plot(x,  y, 'linewidth', 2, 'color', '#abc564'), grid on
xline(1, '-.', {'x = 1'})
legend('logistic fcn', 'scaled reflection', 'skewed fcn')
xlabel('x')

[Alex Sha]

Hi, your data chart looks like peak-type function chart, so just try some typical peak functions, "Asym2Sig" function, shown above, gives more better outcomes.

[Sam Chak]

Hi @Alex Sha, thanks for sharing the information about the Asymmetric Double Sigmoidal function. I don't have Origin Pro, and I have never seen many of those functions before.

https://www.originlab.com/doc/en/Origin-Help/Peak-Analyzer-Functions